\(\int \frac {(a+\frac {b}{x^2})^{5/2}}{x^4} \, dx\) [374]

Optimal result

Integrand size = 15, antiderivative size = 116 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=-\frac {5 a^2 \sqrt {a+\frac {b}{x^2}}}{64 x^3}-\frac {5 a \left (a+\frac {b}{x^2}\right )^{3/2}}{48 x^3}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}-\frac {5 a^3 \sqrt {a+\frac {b}{x^2}}}{128 b x}+\frac {5 a^4 \text {arctanh}\left (\frac {\sqrt {b}}{\sqrt {a+\frac {b}{x^2}} x}\right )}{128 b^{3/2}} \] Output:

-5/64*a^2*(a+b/x^2)^(1/2)/x^3-5/48*a*(a+b/x^2)^(3/2)/x^3-1/8*(a+b/x^2)^(5/ 
2)/x^3-5/128*a^3*(a+b/x^2)^(1/2)/b/x+5/128*a^4*arctanh(b^(1/2)/(a+b/x^2)^( 
1/2)/x)/b^(3/2)
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\frac {\sqrt {a+\frac {b}{x^2}} \left (-\sqrt {b} \left (48 b^3+136 a b^2 x^2+118 a^2 b x^4+15 a^3 x^6\right )+\frac {15 a^4 x^8 \text {arctanh}\left (\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {b+a x^2}}\right )}{384 b^{3/2} x^7} \] Input:

Integrate[(a + b/x^2)^(5/2)/x^4,x]
 

Output:

(Sqrt[a + b/x^2]*(-(Sqrt[b]*(48*b^3 + 136*a*b^2*x^2 + 118*a^2*b*x^4 + 15*a 
^3*x^6)) + (15*a^4*x^8*ArcTanh[Sqrt[b + a*x^2]/Sqrt[b]])/Sqrt[b + a*x^2])) 
/(384*b^(3/2)*x^7)
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {858, 248, 248, 248, 262, 224, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx\)

\(\Big \downarrow \) 858

\(\displaystyle -\int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^2}d\frac {1}{x}\)

\(\Big \downarrow \) 248

\(\displaystyle -\frac {5}{8} a \int \frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{x^2}d\frac {1}{x}-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\)

\(\Big \downarrow \) 248

\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \int \frac {\sqrt {a+\frac {b}{x^2}}}{x^2}d\frac {1}{x}+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\)

\(\Big \downarrow \) 248

\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \int \frac {1}{\sqrt {a+\frac {b}{x^2}} x^2}d\frac {1}{x}+\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\right )+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\)

\(\Big \downarrow \) 262

\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x}-\frac {a \int \frac {1}{\sqrt {a+\frac {b}{x^2}}}d\frac {1}{x}}{2 b}\right )+\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\right )+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x}-\frac {a \int \frac {1}{1-\frac {b}{x^2}}d\frac {1}{\sqrt {a+\frac {b}{x^2}} x}}{2 b}\right )+\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\right )+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {5}{8} a \left (\frac {1}{2} a \left (\frac {1}{4} a \left (\frac {\sqrt {a+\frac {b}{x^2}}}{2 b x}-\frac {a \text {arctanh}\left (\frac {\sqrt {b}}{x \sqrt {a+\frac {b}{x^2}}}\right )}{2 b^{3/2}}\right )+\frac {\sqrt {a+\frac {b}{x^2}}}{4 x^3}\right )+\frac {\left (a+\frac {b}{x^2}\right )^{3/2}}{6 x^3}\right )-\frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{8 x^3}\)

Input:

Int[(a + b/x^2)^(5/2)/x^4,x]
 

Output:

-1/8*(a + b/x^2)^(5/2)/x^3 - (5*a*((a + b/x^2)^(3/2)/(6*x^3) + (a*(Sqrt[a 
+ b/x^2]/(4*x^3) + (a*(Sqrt[a + b/x^2]/(2*b*x) - (a*ArcTanh[Sqrt[b]/(Sqrt[ 
a + b/x^2]*x)])/(2*b^(3/2))))/4))/2))/8
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 248

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) 
  Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ 
p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 858

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + 
b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int 
egerQ[m]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.94

method	result	size
risch	\(-\frac {\left (15 a^{3} x^{6}+118 a^{2} b \,x^{4}+136 a \,b^{2} x^{2}+48 b^{3}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}}{384 x^{7} b}+\frac {5 a^{4} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x}{128 b^{\frac {3}{2}} \sqrt {a \,x^{2}+b}}\)	\(109\)
default	\(\frac {\left (\frac {a \,x^{2}+b}{x^{2}}\right )^{\frac {5}{2}} \left (-3 \left (a \,x^{2}+b \right )^{\frac {5}{2}} a^{4} x^{8}+3 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a^{3} x^{6}-5 \left (a \,x^{2}+b \right )^{\frac {3}{2}} a^{4} b \,x^{8}+15 b^{\frac {5}{2}} \ln \left (\frac {2 b +2 \sqrt {b}\, \sqrt {a \,x^{2}+b}}{x}\right ) a^{4} x^{8}-15 \sqrt {a \,x^{2}+b}\, a^{4} b^{2} x^{8}+2 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a^{2} b \,x^{4}+8 \left (a \,x^{2}+b \right )^{\frac {7}{2}} a \,b^{2} x^{2}-48 \left (a \,x^{2}+b \right )^{\frac {7}{2}} b^{3}\right )}{384 x^{3} \left (a \,x^{2}+b \right )^{\frac {5}{2}} b^{4}}\)	\(186\)

Input:

int((a+b/x^2)^(5/2)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/384*(15*a^3*x^6+118*a^2*b*x^4+136*a*b^2*x^2+48*b^3)/x^7/b*((a*x^2+b)/x^ 
2)^(1/2)+5/128/b^(3/2)*a^4*ln((2*b+2*b^(1/2)*(a*x^2+b)^(1/2))/x)*((a*x^2+b 
)/x^2)^(1/2)*x/(a*x^2+b)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.72 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\left [\frac {15 \, a^{4} \sqrt {b} x^{7} \log \left (-\frac {a x^{2} + 2 \, \sqrt {b} x \sqrt {\frac {a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ) - 2 \, {\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{768 \, b^{2} x^{7}}, -\frac {15 \, a^{4} \sqrt {-b} x^{7} \arctan \left (\frac {\sqrt {-b} x \sqrt {\frac {a x^{2} + b}{x^{2}}}}{b}\right ) + {\left (15 \, a^{3} b x^{6} + 118 \, a^{2} b^{2} x^{4} + 136 \, a b^{3} x^{2} + 48 \, b^{4}\right )} \sqrt {\frac {a x^{2} + b}{x^{2}}}}{384 \, b^{2} x^{7}}\right ] \] Input:

integrate((a+b/x^2)^(5/2)/x^4,x, algorithm="fricas")
 

Output:

[1/768*(15*a^4*sqrt(b)*x^7*log(-(a*x^2 + 2*sqrt(b)*x*sqrt((a*x^2 + b)/x^2) 
 + 2*b)/x^2) - 2*(15*a^3*b*x^6 + 118*a^2*b^2*x^4 + 136*a*b^3*x^2 + 48*b^4) 
*sqrt((a*x^2 + b)/x^2))/(b^2*x^7), -1/384*(15*a^4*sqrt(-b)*x^7*arctan(sqrt 
(-b)*x*sqrt((a*x^2 + b)/x^2)/b) + (15*a^3*b*x^6 + 118*a^2*b^2*x^4 + 136*a* 
b^3*x^2 + 48*b^4)*sqrt((a*x^2 + b)/x^2))/(b^2*x^7)]
 

Sympy [A] (verification not implemented)

Time = 7.62 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=- \frac {5 a^{\frac {7}{2}}}{128 b x \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {133 a^{\frac {5}{2}}}{384 x^{3} \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {127 a^{\frac {3}{2}} b}{192 x^{5} \sqrt {1 + \frac {b}{a x^{2}}}} - \frac {23 \sqrt {a} b^{2}}{48 x^{7} \sqrt {1 + \frac {b}{a x^{2}}}} + \frac {5 a^{4} \operatorname {asinh}{\left (\frac {\sqrt {b}}{\sqrt {a} x} \right )}}{128 b^{\frac {3}{2}}} - \frac {b^{3}}{8 \sqrt {a} x^{9} \sqrt {1 + \frac {b}{a x^{2}}}} \] Input:

integrate((a+b/x**2)**(5/2)/x**4,x)
 

Output:

-5*a**(7/2)/(128*b*x*sqrt(1 + b/(a*x**2))) - 133*a**(5/2)/(384*x**3*sqrt(1 
 + b/(a*x**2))) - 127*a**(3/2)*b/(192*x**5*sqrt(1 + b/(a*x**2))) - 23*sqrt 
(a)*b**2/(48*x**7*sqrt(1 + b/(a*x**2))) + 5*a**4*asinh(sqrt(b)/(sqrt(a)*x) 
)/(128*b**(3/2)) - b**3/(8*sqrt(a)*x**9*sqrt(1 + b/(a*x**2)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (92) = 184\).

Time = 0.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.64 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=-\frac {5 \, a^{4} \log \left (\frac {\sqrt {a + \frac {b}{x^{2}}} x - \sqrt {b}}{\sqrt {a + \frac {b}{x^{2}}} x + \sqrt {b}}\right )}{256 \, b^{\frac {3}{2}}} - \frac {15 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {7}{2}} a^{4} x^{7} + 73 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {5}{2}} a^{4} b x^{5} - 55 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {3}{2}} a^{4} b^{2} x^{3} + 15 \, \sqrt {a + \frac {b}{x^{2}}} a^{4} b^{3} x}{384 \, {\left ({\left (a + \frac {b}{x^{2}}\right )}^{4} b x^{8} - 4 \, {\left (a + \frac {b}{x^{2}}\right )}^{3} b^{2} x^{6} + 6 \, {\left (a + \frac {b}{x^{2}}\right )}^{2} b^{3} x^{4} - 4 \, {\left (a + \frac {b}{x^{2}}\right )} b^{4} x^{2} + b^{5}\right )}} \] Input:

integrate((a+b/x^2)^(5/2)/x^4,x, algorithm="maxima")
 

Output:

-5/256*a^4*log((sqrt(a + b/x^2)*x - sqrt(b))/(sqrt(a + b/x^2)*x + sqrt(b)) 
)/b^(3/2) - 1/384*(15*(a + b/x^2)^(7/2)*a^4*x^7 + 73*(a + b/x^2)^(5/2)*a^4 
*b*x^5 - 55*(a + b/x^2)^(3/2)*a^4*b^2*x^3 + 15*sqrt(a + b/x^2)*a^4*b^3*x)/ 
((a + b/x^2)^4*b*x^8 - 4*(a + b/x^2)^3*b^2*x^6 + 6*(a + b/x^2)^2*b^3*x^4 - 
 4*(a + b/x^2)*b^4*x^2 + b^5)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.03 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=-\frac {\frac {15 \, a^{5} \arctan \left (\frac {\sqrt {a x^{2} + b}}{\sqrt {-b}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-b} b} + \frac {15 \, {\left (a x^{2} + b\right )}^{\frac {7}{2}} a^{5} \mathrm {sgn}\left (x\right ) + 73 \, {\left (a x^{2} + b\right )}^{\frac {5}{2}} a^{5} b \mathrm {sgn}\left (x\right ) - 55 \, {\left (a x^{2} + b\right )}^{\frac {3}{2}} a^{5} b^{2} \mathrm {sgn}\left (x\right ) + 15 \, \sqrt {a x^{2} + b} a^{5} b^{3} \mathrm {sgn}\left (x\right )}{a^{4} b x^{8}}}{384 \, a} \] Input:

integrate((a+b/x^2)^(5/2)/x^4,x, algorithm="giac")
 

Output:

-1/384*(15*a^5*arctan(sqrt(a*x^2 + b)/sqrt(-b))*sgn(x)/(sqrt(-b)*b) + (15* 
(a*x^2 + b)^(7/2)*a^5*sgn(x) + 73*(a*x^2 + b)^(5/2)*a^5*b*sgn(x) - 55*(a*x 
^2 + b)^(3/2)*a^5*b^2*sgn(x) + 15*sqrt(a*x^2 + b)*a^5*b^3*sgn(x))/(a^4*b*x 
^8))/a
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\int \frac {{\left (a+\frac {b}{x^2}\right )}^{5/2}}{x^4} \,d x \] Input:

int((a + b/x^2)^(5/2)/x^4,x)
 

Output:

int((a + b/x^2)^(5/2)/x^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.20 \[ \int \frac {\left (a+\frac {b}{x^2}\right )^{5/2}}{x^4} \, dx=\frac {-15 \sqrt {a \,x^{2}+b}\, a^{3} b \,x^{6}-118 \sqrt {a \,x^{2}+b}\, a^{2} b^{2} x^{4}-136 \sqrt {a \,x^{2}+b}\, a \,b^{3} x^{2}-48 \sqrt {a \,x^{2}+b}\, b^{4}-15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x -\sqrt {b}}{\sqrt {b}}\right ) a^{4} x^{8}+15 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {a \,x^{2}+b}+\sqrt {a}\, x +\sqrt {b}}{\sqrt {b}}\right ) a^{4} x^{8}}{384 b^{2} x^{8}} \] Input:

int((a+b/x^2)^(5/2)/x^4,x)
 

Output:

( - 15*sqrt(a*x**2 + b)*a**3*b*x**6 - 118*sqrt(a*x**2 + b)*a**2*b**2*x**4 
- 136*sqrt(a*x**2 + b)*a*b**3*x**2 - 48*sqrt(a*x**2 + b)*b**4 - 15*sqrt(b) 
*log((sqrt(a*x**2 + b) + sqrt(a)*x - sqrt(b))/sqrt(b))*a**4*x**8 + 15*sqrt 
(b)*log((sqrt(a*x**2 + b) + sqrt(a)*x + sqrt(b))/sqrt(b))*a**4*x**8)/(384* 
b**2*x**8)